Total Derivatives and Total Differential

Question

I am confused between total derivatives and total differential. What is the difference between total derivatives and total differential?

For a question like this, you should provide a definition of each of these terms to get a good answer. — Bernard W, Jul 6 '17 at 1:04

Doug M · Accepted Answer · 2017-07-06 01:48:03Z

1

the total differential is

$dz = \frac {\partial z}{\partial x} dx + \frac {\partial z}{\partial y} dy$

How much do we expect $z$ to change for some changes in $x$ and $y?$

The total derivative is

$\frac {dz}{dt} = \frac {\partial z}{\partial x} \frac {dx}{dt} + \frac {\partial z}{\partial y} \frac {dy}{dt}$

$x,y$ are both functions with respect to parameter $t$ what is the derivative of $z$ with respect to this parameter?

answered Jul 6 '17 at 1:48

Doug M

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$\begingroup$ Notice that in the equation for the total derivative, two different functions are both being called $z$. This is a common abuse of notation, but I think it sometimes causes confusion. $\endgroup$ – littleO Jul 6 '17 at 2:49

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Cauchy · Accepted Answer · 2017-07-06 01:05:48Z

Let $f: U \subset \Bbb R^n \to \Bbb R^m$ be differentiable.

The total derivative of $f$ at $a$ is the linear map $df_a$ such that $f(a+t) - f(a) = df_a(t) + o(t)$.

For $m=1$, the total differential of $f$ is

$$df = \sum_{i=1}^m \frac{\partial f}{\partial x_i} dx_i$$

Hope this helps.

2 Answers 2